Wear and Contact Mechanics

247 Pages·2014·22.405 MB·English

by  Luis Rodriguez-tembleque, Ferri Aliabadi

Wear and
Contact Mechanics
Edited by
Luis Rodriguez-Tembleque
Ferri Aliabadi
Wear and Contact Mechanics
Special topic volume with invited peer reviewed papers only.
Edited by
Luis Rodriguez-Tembleque and Ferri Aliabadi
Copyright  2014 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this publication may be reproduced or
transmitted in any form or by any means without the written permission of the
publisher.

Trans Tech Publications Ltd
Kreuzstrasse 10
CH-8635 Durnten-Zurich
Switzerland
http://www.ttp.net

Volume 618 of
Key Engineering Materials
ISSN print 1013-9826
ISSN cd 1662-9809
ISSN web 1662-9795

Full text available online at http://www.scientific.net

Distributed worldwide by and in the Americas by
Trans Tech Publications Ltd Trans Tech Publications Inc.
Kreuzstrasse 10 PO Box 699, May Street
CH-8635 Durnten-Zurich Enfield, NH 03748
Switzerland USA
Phone: [phone]
Fax: [phone] Fax: [phone]
e-mail: [email protected] e-mail: [email protected]
Preface

This special topic volume is a compilation of works contributed by experts from
the international scientific community in the field of Wear and Contact
Mechanics.
This volume presents ten papers that cover different aspects of the current
areas of research in Wear and Contact Mechanics using new innovative
theoretical and computational approaches based on the Finite Element Methods
(FEM) and/or the Boundary Element Methods (BEM). A number of topics are
addressed, such as anisotropic contact, anisotropic wear, crack-face contact
and semi-permeable electric/magnetic boundary conditions in multifield
materials (e.i. piezoelectric solids), dynamic contact modeling, fatigue crack
growth life estimation, fatigue life assessment, fretting fatigue, fretting wear
computing, rolling contact modeling, or treatment of nonmatching interfaces in
contact problems.
The editors would like to thank the contributors of papers, the reviewers and the
Key Engineering Materials Editorial Board for helping put together this special
issue. Finally, we would like to dedicate this work to the memory of Prof. Ramón
Abascal (1956-2013). He was a great researcher in this topic and, above all, a
good person and friend.

Editors

Luis Rodriguez-Tembleque
M.H. Aliabadi
Prologue

Ramón A. Abascal García
(1956 – 2013)
Prof. Ramón Abascal was full professor of Continuum Mechanics and Theory of
Structures at the “Escuela Técnica Superior de Ingeniería” (ETSI) of the
University of Seville, Spain. He was born in Seville in 1956 and graduated in
mechanical engineering at ETSI in 1979. He received his Ph.D. from University
of Seville five years later and in 1986 started his teaching career as associated
professor at the Department of Continuum Mechanics and Theory of Structures
of the ETSI, becoming full professor in 1995.
His research interests were focused on the Boundary Element Method and its
applications, with emphasis on elastodynamics (seismic wave propagation and
scattering in non-homogenous viscoelastic soils, seismic response of
foundations including dynamic soil-structure interaction and nonlinear contact
effects due to uplift, guided wave scattering, and ultrasonic waves), fracture
mechanics, contact problems (including friction, rolling and wear) and
substructure coupling techniques using Lagrange multipliers. During his
scientific career, he advised three Ph.D. students and published over forty
papers in the most prestigious scientific journals.
But his dedication was not only restricted to research; he was also very
concerned with the education of future engineers. He taught with excellence
different subjects, such as: Theory of Structures, Advanced Analysis of
Structures, Steel Structures and Advanced Finite Elements. Due to his
commitment to academic excellence, Ramón was appointed Head of Studies at
the ETSI from 2006 to 2010.
First as a Ph.D. student and later as a colleague, I have witnessed the
enormous rigor, high demand and passion that Ramón projected on all his
work. Everybody in the scientific and educational community, who were
fortunate to know him, will remember his enormous generosity, his sense of
humor, and, most of all, his significant scientific work. Colleagues and friends
will never forget Ramón and will deeply regret losing him.
This work is dedicated to the memory of Prof. Ramón Abascal. We mourn his
untimely death as we lose a great engineering educator, researcher and, above
all, a good person and friend.

Luis Rodriguez-Tembleque
Table of Contents
Preface and Prologue
The Influence of Equivalent Contact Area Computation in Extended Node to Surface
Contact Elements
G. Wautelet and J.P. Ponthot 1
A Partitioned Formulation for FEM/BEM Coupling in Contact Problems Using Localized
Lagrange Multipliers
J.A. González, K.C. Park and R. Abascal 23
On Steady Wear States for Monotonic Relative Sliding of Contacting Bodies
I. Páczelt and Z. Mróz 49
Anisotropic Contact and Wear Simulation Using Boundary Elements
L. Rodríguez-Tembleque, M.H. Aliabadi and R. Abascal 73
Life Assessment in Fretting Fatigue
C. Navarro, J. Vázquez and J. Domínguez 99
Transient Dynamic Analysis of Cracked Multifield Solids with Consideration of Crack-Face
Contact and Semi-Permeable Electric/Magnetic Boundary Conditions
M. Wünsche, A. Sáez, F. García-Sánchez, C.Z. Zhang and J. Domínguez 123
BEM and Tangent Operator Technique Applied to Analysis of Contact Problems
S.G. Ferreira Cordeiro and E.D. Leonel 151
Effect of Friction on the Size of the Near-Tip Contact Zone in a Penny-Shaped Interface
Crack
E. Graciani, V. Mantič and F. París 179
Closed-Form Solution of the Frictional Sliding Contact Problem for an Orthotropic Elastic
Half-Plane Indented by a Wedge-Shaped Punch
A. Kucuksucu, M.A. Guler and A. Avci 203
Nonlinear Time Spectral Analysis for a Dynamic Contact Model with Buckling for an
Elastic Plate
A.D. Muradova and G.E. Stavroulakis 227
Key Engineering Materials Vol. 618 (2014) pp 1-22
© (2014) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/KEM.618.1
The influence of equivalent contact area computation
in extended node to surface contact elements
WAUTELET Gae¨tan1;a, PONTHOT Jean-Philippe1;b
LTAS-MN2L - Non-Linear Computational Mechanics, Departement of Aerospace & Mechanical
Engineering, University of Liège, Chemin des chevreuils Bat B52/3, 4000 Liège, Belgium
{agwautelet, bJP.Ponthot } @ulg.ac.be
Keywords: Contact mechanics, Finite element method, Penalty method, Node to surface, Friction-
less, Area regularization, Axisymmetric.
Abstract. This article aims at extending the node to surface formulation for contact problems with
an area regularization as proposed by [1]. For that purpose, two methods are proposed to compute
the equivalent contact area attributed to each slave node. The first method, which is based on a geo-
metrical approach through force equivalence, is an original extension of the one proposed in [1] for
two-dimensional contact problems, i.e. plane stress and plane strain state, to the axisymmetric mod-
elling context. The second method relies on an energy consistent way obtained through the virtual
work principle and the same expression for the equivalent contact area as the one originally cited in
[2]isthenrecovered.First,thenodetosurfacestrategywitharearegularizationisintroducedandthe
aforementioned methods for the equivalent contact area are presented in detail and compared. After-
wards a consistent linearization technique is applied to achieve a quadratic convergence rate in the
NewtonRaphsoniterativeprocedureusedtosolvethenon-linearequilibriumequationsoftheunder-
lying finite element model. Finally, two axisymmetric numerical examples are provided in order to
comparetheaforementionedequivalentcontactareaevaluationsandtodemonstratetheperformance
andtherobustnessoftheconsistentapproachespeciallyintheneighbourhoodtherevolutionaxis.
Introduction
Nowadays, the node to surface formulation [3, 4] is widespread in commercial finite element codes
anditiscommonlyusedtosolvethecontactproblemsinlargedeformationwhichrangefromthesheet
metal forming processes to the impact tests in crashworthiness assessment. Of particular interest in
practical engineering applications is the knowledge of the contact pressure and frictional shear stress
distributionacrossthecontactingsurfaces.However,intheclassicalnodetosurfaceformulation,there
areonlycontactforcesdirectlyavailableatthenodesincontactbutsomepost-processingtechniques
maybeusedtotransformthemintocontactpressureandfrictionalshearstress[5].Fromatribological
pointofview,micro-macroapproacheshaverecentlyproposedadvancedcontactmechanicalconstitu-
tivemodelsintermsofcontactpressure[6,7,8]andfrictionalshearstress[9,10,2].Henceforth,their
applicability requires to include an area term in the classical node to surface formulation, if one does
not want to rely on more advanced contact formulations, such as mortar-like [11, 12, 13] or segment
to segment [14, 15, 16] strategies, where a contact pressure and frictional shear stresses are naturally
availableintheweakformassociatedwiththecontactproblem.
Historically, Zavarise et al. [6] introduced for the first time the idea to associate an equivalent
contact area to the nodes of the slave surface in a thermomechanical two-dimensional contact finite
element formulation due to the presence of complex constitutive models for the thermal contact re-
sistance. These authors decided to compute this area on the slave surface for simplicity’s sake, even
ifthe master area is immediately available in the node to surface formulation, because the number of
slave nodes in contact with the same master segment should be known in order to determine the area
partonthemastersegmentonwhicheachslavenodeacts.
2 Wear and Contact Mechanics
Following this former work focused on the thermal treatment of the contact formulation, Stup-
kiewicz [2] proposed an extended node to surface contact element dedicated to more complex fric-
tional laws, depending on the contacting surface area stretching, than the classical Coulomb’s law in
two-dimensional and axisymmetric modelling context. Since the equivalent contact area term gives
risetoacouplingbetweenaslavenodeanditsneighbouringnodescalledauxiliaryslavenodes,their
degrees of freedom should be taken into account to derive the full tangent stiffness matrix associated
to this contact element. Even in the frictionless case, the equivalent contact area makes the tangent
stiffness matrix unsymmetric and the quadratic convergence rate is reached in the Newton-Raphson
iterativescheme,justincaseofconsistentlinearization,asexpected.
Recently, Zaravise and De Lorenzis [1] revisited the contact patch test with a frictionless node to
surface formulation. If the penalty method is adopted, the non-penetration constraint is regularized
withoutnoadditionalunknownsandapenaltycoefficientisintroducedtocontroltheamountofpen-
etration between the contacting bodies. Contrary to Lagrange multiplier method, the non-penetration
constraint is no longer exactly satisfied but the exact solution to the contact problem is theoretically
recoveredwhenthepenaltycoefficienttendstoinfinity.Nevertheless,inreal-lifeapplications,afinite
value of penalty coefficient is set depending on the problem upon consideration, and it is generally
boundedtoavoidill-conditioningoftheglobalsystemofequilibriumequations.
The contact patch test assesses the ability of a contact formulation to transfer a constant con-
tact pressure through two deformable contacting bodies regardless their discretization [17], so that a
constant penetration is expected with the penalty method. It is well known that the node to surface
formulation is not able to pass the contact patch test. Therefore, local solution errors may occur and
won’tdisappearwithmeshrefinement.Inconjunctionwiththepenaltymethod,thecontactalgorithm
treatmentismuchtrickierduetothepresenceofnon-zeropenetrationofthecontactingbody.Tothat
purpose, the authors proposed successive improvements of the contact algorithm towards a node to
surfaceformulationpassingthepatchtestforlinearfiniteelements.Accordingtothem,ifalltheslave
nodes are projected against the same master segment, which is a special case encountered by their
algorithm,thepresenceofanequivalentcontactareaisasufficientconditiontopassthecontactpatch
test.
In fact, the finite element method converts a pressure distribution applied on a arbitrary two-
dimensional surface into uneven nodal forces distribution. In the particular case where the pressure
distribution is constant, the nodal force for the nodes at the boundary of the surface is equal to half
of the force attributed to each node in the interior of the surface. Since the contact force is linearly
related to the penetration by means of a penalty coefficient, considered as a constant value for each
slave node, the classical node to surface formulation is not able to reproduce a constant penetration.
To solve the problem, the penalty coefficient is weighted by an equivalent contact area, in order to
obtain a contact pressure immediately available at the slave node instead of a contact force on one
hand, and on the other hand to be able to reproduce a constant contact pressure over the contacting
surface,ifthisareaisequaltothesumofhalfthelengthofthecoincidentsegmentstoeachslavenode.
Themotivationofthepresentpaperistoextendtheaforementionedworktotheaxisymmetricmod-
ellingcontextwithlinearfiniteelementsandtodemonstratethattheequivalentcontactareashouldbe
carefullycomputedinordertobeabletoreproduceaconstantcontactpressurewhichisthefirststep
towardsthedevelopmentofanodetosurfaceformulationpassingthepatchtest.Forthatpurpose,this
paperisorganizedintotwomainpartsasfollows.Thefirstpartoutlinesthenodetosurfacewitharea
regularization as described in [1, 2], and discussed in details the two computational methods used to
determine the equivalent contact area in the axisymmetric modelling context, as well as their contri-
bution to the full tangent stiffness matrix through a consistent linearization process. The second part